materials

Article

Dynamic Ferromagnetic Hysteresis Modelling Usinga Preisach-Recurrent Neural Network Model

Christian Grech 1,2,* , Marco Buzio 2 , Mariano Pentella 2,3 and Nicholas Sammut 1,2

1 Faculty of Information and Communications Technology, University of Malta, MSD2080 Msida, Malta;nicholas.sammut@um.edu.mt

2 CERN, European Organization for Nuclear Research, 1211 Geneva, Switzerland;Marco.Buzio@cern.ch (M.B.); mariano.pentella@cern.ch (M.P.)

3 Department of Applied Science and Technology, Politecnico di Torino, 10129 Turin, Italy* Correspondence: christian.grech.12@um.edu.mt

Received: 10 May 2020; Accepted: 29 May 2020 ; Published: 4 June 2020�����������������

Abstract: In this work, a Preisach-recurrent neural network model is proposed to predict the dynamichysteresis in ARMCO pure iron, an important soft magnetic material in particle accelerator magnets.A recurrent neural network coupled with Preisach play operators is proposed, along with a novelvalidation method for the identification of the model’s parameters. The proposed model is foundto predict the magnetic flux density of ARMCO pure iron with a Normalised Root Mean SquareError (NRMSE) better than 0.7%, when trained with just six different hysteresis loops. The modelis evaluated using ramp-rates not used in the training procedure, which shows the ability of themodel to predict data which has not been measured. The results demonstrate that the Preisach modelbased on a recurrent neural network can accurately describe ferromagnetic dynamic hysteresis whentrained with a limited amount of data, showing the model’s potential in the field of materials science.

Keywords: ARMCO pure iron; dynamic hysteresis loop; machine learning; magnetic properties;particle accelerators; Preisach; recurrent neural networks

1. Introduction

The phenomenon of hysteresis occurs when a system is not only dependent on present inputvalues, but also on past input values. The resulting hysteresis loops make modelling a challengedue to the non-linearity exhibited. In phenomena such as ferromagnetism, dynamic effects adda further aspect of complexity. The interaction between electric and magnetic fields results ininduced eddy currents, which modify the magnetic hysteresis characteristics. Proposed modelsto predict hysteresis can be classified into two main categories: physical models and phenomenologicalmodels. The physical models are built on the description of the object being modelled usingphysics laws. Unfortunately, in most cases such models are mathematically complex and requiredetailed knowledge of the physical properties of the material being modelled. Phenomenologicalmodels make use of conventional identification methods, which typically have no physical meaning.Such models are mostly empirical, based on previously acquired experimental data. The most popularphenomenological hysteresis model is the Preisach model [1], used in a vast range of applications.Among the different hysteresis models available in literature, the Preisach-type models prove to havea great potential to explain various magnetization processes, and is one of the most popular models tocapture the hysteresis behaviour in non-linear systems.

The biggest challenge in implementing a Preisach model is the identification of the modelparameters, because the use of empirical methods makes the procedure problem-dependent [2].There are numerous identification methods used in literature [3–5]. In most cases, such a function

Materials 2020, 13, 2561; doi:10.3390/ma13112561 www.mdpi.com/journal/materials

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can be identified using measured data and applying a numerical approach, which can be a specificformulation [6,7] or using conventional methods such as an Everett integral [8] or Gaussian/Lorentzianfunctions [9]. The function can then be applied in software using a look-up table of values.Other ‘black-box’ identification techniques used alongside the Preisach model include geneticalgorithms [10], fuzzy models [11] and artificial neural networks (ANN) [12–14]. In [15], Saliah et al.showed that using an ANN can match results produced by the conventional methods, considerablyreducing the time overhead. In [16], Serpico and Visone build an ANN hysteresis model, which is ableto model rate-independent hysteresis when combined with Preisach operators as inputs. Similarly,different neural network configurations have been used for modelling the rate-independent hysteresisof magnetic shape memory alloys [17,18].

The Preisach model implementation described above is rate-independent, meaning that thehysteresis output is determined solely by the input’s extreme values, and the input ramp rates do notimpact the hysteresis loop. Mayergoyz [19] introduced the dependence of the weight function on thespeed of output variations; similarly, Mrad and Hu [20] and Song and Li [21] proposed an input-ratedependence of the weight function. Both methods suppose the weight function is the right place toadd in dynamic behaviors. On the other hand, in [22] a linear dynamic model is added before theclassical Preisach operator and the dynamics are assumed to only happen inside the linear dynamicpart. The latter cascade structure can be referred to as an ‘external dynamic hysteresis model’ and isfound in several works [23–25]. In this work, we will consider the rate-dependency without includingan additional element. This can be accounted for by implementing the Preisach model’s weightingfunction using dynamic neural networks such as a recurrent neural network (RNN), where each layerhas a recurrent connection, allowing the network to have an infinite dynamic response to time seriesinput data. An example of the implementation of such model configurations for hysteretic data can befound in [26], where an Elman RNN was successful in predicting the major loop at several frequencies.In [27], with the addition of Preisach operators at the input stage of an internal time-delay neuralnetwork, the major and minor loop hysteresis of a Giant Magnetostrictive Actuator was modelledsuccessfully, however this work did not demonstrate functionality in the saturation regions.

In this paper, a Preisach model is implemented using a single RNN, which is able to predict thedifferent dynamic hysteresis loops of ferromagnetic materials, when a limited amount of measurementdata is available. In particular, the model is trained using three particular frequencies and tested on adata set which consists of a different frequency. As a novel contribution to the current research state,both major and minor loop hysteresis are investigated, including saturation regions. The motivationbehind this work lies in the prediction of the dynamic behaviour of materials used in the manufacturingof magnets for particle accelerators. As an example, ARMCO is used as a test material being usedas part of the superconducting magnets for the High Luminosity upgrade of the Large HadronCollider at the European Nuclear Research Centre (CERN) [28,29], where pulsed fields are employed.Hence, the knowledge of the material’s dynamic behaviour, which is not easily modelled, is requiredand this work attempts to propose an accurate model, trained with a limited amount of measurementdata, which can also be used at a moderate computational cost. Whilst a vector hysteresis modelis required for a 3D model of an accelerator magnet, the model can be used to describe the verticalfield used for the RF control of synchrotrons. The description of the eddy-currents transients whenramping up the material during a test is complicated due to its hysteretic behaviour and the geometry,generally toroidal, which complicates the formulation [30]. This combined with the intrinsic nature ofthe flux-metric method adopted for the material measurement, introduces an uncertainty componenton the results, especially on the coercive field determination [31,32]. The knowledge of the dynamicbehaviour of the material can be potentially used to extrapolate the data in DC, allowing the separationof the rate-independent hysteresis from the rate-dependent part, and reducing the overall uncertainty.

In Section 2, the experimental details behind the measurements, the theory behind the model andthe validation technique are described. Section 3 describes the results obtained, including the resultsof a univariate sensitivity analysis. As test material, ARMCO was considered, being an important

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yoke material in particle accelerators [33]. Moreover, having an electrical conductivitiy in the orderof 108 S/m and a relative permeability in ther order of 3000, the effect of the eddy currents on thehysteresis loop can be measured with a good signal-to-noise ratio.

2. Materials and Methods

2.1. Experimental Details

The measurements in this work are performed using a split-coil permeameter [28], shown inFigure 1, and performed on toroidal test specimens. The equipment consists of three 90-turn coils,separable by means of an opening mechanism. A slot allows the insertion of the sample to be tested inthe permeameter, therefore avoiding the time-demanding operation of a custom coil winding ontothe sample [34]. The two outermost coils are used as excitation coils and powered in series up to amaximum current of 40 A, corresponding to a maximum magnetic field of 24 kAm−1. The innermostcoil is used to detect the induced voltage. The entire system is cooled by compressed air.

Figure 1. The split-coil permeameter.

The measurements are performed by means of the fluxmetric method, as shown in Figure 2.The test specimen is magnetized by the two excitation coils, having in total Ne = 180 turns and poweredin series by a voltage-controlled current generator. Given r1 and r2 respectively the inner and the outerradius of the test specimen, the magnetic field is equal to:

H(t) =Nei(t)2πr0

(1)

where:r0 =

r2 − r1

ln r2r1

(2)

The magnetic flux density is evaluated by:

B(t) =1

As

(Φ(t)Ns− µ0H(At − As)

)(3)

where As is the cross-sectional area of the sample, At the cross-sectional area of the sensing coil, Ns isthe number of turns of the sensing coil, µ0 the permeability of the free space and Φ(t) the magneticflux, evaluated by integrating the induced voltage.

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D

V

ADC

PC

DAC

Digital acquisition

system

PermeameterExcitation coilSensing

coil

A

Voltage controlled current generator

Figure 2. Measurement system layout.

Both the current and the voltage are acquired by a digital acquisition system (DAQ), a NI 4461 [35]by National Instruments at frequency of 20 kHz. In particular, the value of the current is acquired usinga MACC2 PLUS direct current transducer [36]. The value of the magnetic flux density is acquiredwith an uncertainty of 0.5 mT whereas the magnetic field is known with an accuracy of 0.1 Am−1.The current is ramped back and forth, between positive and negative symmetric values, and at the endof each ramp the current is kept constant for 0.5 s.

For the major hysteresis loops, the plateau amplitudes are chosen in such a way that the materialis brought in saturation state. In order to acquire only the hysteresis loop without including the initialmagnetization branch, a pre-cycle is applied before the specific cycle. In the case of the minor loops,the same procedure is carried out, with the difference that the plateau amplitudes at each cycle areincreased. Between each acquisition and the following one, the sample is demagnetized. The magneticflux density, B(t) in different dynamic hysteretic conditions obtained using the magnetic field, H(t) arethe source of information for this work. In the case of minor loops, this work is limited to symmetrical,first-order curves.

2.2. Hysteresis Modelling Based on Preisach Memory

In general, the Preisach model is expressed using a double integrator in continuous form as [1]:

y(t) =∫ ∫α≥β

µ(α, β)γαβu(t)dαdβ (4)

where y(t) is the model output at time t, u(t) is the model input at time t, while γαβ are elementaryrectangular hysteresis operators with α and β being the up and down switching values, respectively.These operators can only assume a value of +1 or −1. The density function µ(α, β) is a weightingfunction, which represents the only model unknown which has to be determined from experimentaldata. In [37,38], following a change in co-ordinates r = (α− β)/2, v = (α + β), µ = µ(v + r, v− r), it isshown that the boundary between the +1 and −1 regions in the Preisach half-plane with coordinatesr > 0, v ∈ R, is described by the function v = P[u](t), known as the play operator. This makes itpossible to rearrange Equation (4) as:

y(t) =∫ +∞

0g(r, P[u(t)]) dr (5)

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which can be discretized to n play operators as follows:

y(t) =n

∑j=1

φjPj[u](t) (6)

where φj represents the density function of the jth play operator, which has to be identified. The playoperator is shown in Figure 3, and defined in Equation (7):

Pj[u](t) = max(u(t)− rj, min(u(t) + rj, Pj[u](t− 1))) (7)

Pj[0] = max(u(0)− rj, min(u(0) + rj, k0)) (8)

where k0 is the initial condition of the operator and rj represents the memory depth as follows:

rj =j− 1

n[max(u(t))−min(u(t))] (9)

where j = 1, 2, 3, . . . n.

Figure 3. Play operator.

2.3. Identifying φ Using Recurrent Neural Networks

Artificial neural networks are able to map non-linear data in various applications. In this case,a recurrent neural network (RNN) will be used to replace the density function φj of the discrete Preisachmodel (Equation (6)), as these structures are recognized for their ability to model any non-lineardynamic system, up to a given degree of accuracy [39]. RNNs are distinguished from feed-forwardnetworks by the feedback loop connected to their past decisions, ingesting their own outputs momentafter moment as input. This means that such networks can be used to model dynamic characteristics.Sequential information is preserved in the recurrent network’s context layer, which manages to spanmany time steps as it cascades forward to affect the processing of each new example.

In general, a RNN can be seen as a group of nodes, consisting of three different kinds of nodes,namely the input, hidden and output nodes, organized in separate layers, as shown in Figure 4.While the input and output layers consist of feed-forward connections, the hidden layer has recurrentones. At each time step, t, the input vector v(t) is processed at the input layer. Each instant v(t)is summed with the bias vector 1b and multiplied by the input weight matrix 1w. Analogously,the internal state z(t), delayed by a number of time instants d, is multiplied by the gain factor hw andadded to the input state as follows:

z(t) = fh[1w(v(t) + 1b) + hw(z(t− d))] (10)

where fh(x) is the activation function, in this case a hyperbolic tangent function, which is given as:

fh(x) =2

1 + e−2x − 1 (11)

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The internal state z(t) is then added with bias 2b, multiplied by the weight 2w, and the result is passedthrough a linear activation function fo(x) as follows:

y(t) = fo[2w(z(t) + 2b)] (12)

where y(t) is the predicted output at time t.

Figure 4. Schematic diagram of the Preisach-RNN model.

The Deep Learning Toolbox [40] by MATLAB is used to determine the weights of the networkusing the layrecnet command. The training algorithm is the Levenberg-Marquadt algorithm [41],which is a non-linear least squares optimization algorithm incorporated into the backpropagationalgorithm for training neural networks as demonstrated in detail in [42]. The algorithm aims tooptimize the weights according to the following objective function:

V(t) =12(y(t)− y(t))T(y(t)− y(t)) (13)

which leads to the update of the weights by the following formula:

iw(t + 1) = iw(t) + η

(−∂V

∂t

)(14)

where η is a positive number representing the learning rate of the weights.

2.4. Data Analysis Approach

The input data to the model include the magnetic field H(t), its derivative H(t) and a number ofplay operators Pj[H](t) (chosen according to the complexity of the data). All data is normalised to therange of the minor loop data, such that data is approximately in the range [−1, 1]. The play operatoroutputs, Pj, are subsequently calculated based on the normalised magnetic field signal. In this work,the predicted variable is the magnetic flux density measured in Section 2.1, and all measurements aredownsampled to 200 Hz.

The data set used for training, validation and testing comprises of three minor loops and threemajor loops ramping at 1025, 1554 and 6135 Am−1s−1, as shown in Figure 5. This data is divided intothree subsets in an interleaved manner. The first subset is the training set (70% of the data), which isused for updating the network weights and biases. The second subset is the validation set (15% ofthe data), used to decide when to stop training the model, whilst the final set is the testing set (15% ofthe data), which is used to select the best model structure. Once the network is trained, and the bestmodel structure is chosen, an evaluation signal is used to demonstrate the performance of the model.

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The evaluation data set consists of three major loops ramping at 3067 Am−1s−1 and one minor loopramping at various random ramp-rates between 1000 and 6200 Am−1s−1, as shown in Figure 6.

0 50 100 150 200 250

Time [s]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

No

rma

lise

d in

pu

ts

Figure 5. Magnetic field data and its derivative used for training, validation and testing the model.

0 10 20 30 40 50 60 70

Time [s]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

No

rma

lise

d in

pu

ts

Figure 6. Magnetic field data and its derivative used for evaluating the model.

2.5. Model Validation

Model validation is the process of choosing the best model parameters and making sure that themodel is robust to new data. The optimal number of nodes in the hidden layer, h is chosen by doing asearch over a range of values. The complete validation process explained below consists of three loops,and is represented by a flowchart in Figure 7.

In the innermost loop, each neural network is trained using the training set as described inSection 2.3 over a certain number of repetitions called epochs. The number of epochs is determinedusing a method called early stopping [43,44]. In this technique, the error on the validation set ismonitored during the training process. The validation error normally decreases during the initialphase of training, as does the training set error. However, when the network begins to overfit thedata, the error on the validation set typically begins to rise. When the validation error increases for aspecified number of iterations, the training is stopped, and the weights and biases at the minimumof the validation error are returned. Finally, the testing set is used to obtain the performance of theparticular network.

Each time a neural network is trained, a different solution is obtained due to different and randominitial weight and bias values. As a result, different neural networks trained on the same problem cangive different outputs for the same input. To ensure that a neural network of good accuracy has beenfound, each network is retrained for a number of times, N, which in this work is 10.

In the outermost loop, the number of hidden nodes is varied over the range starting from hmin tohmax with increments of dh. The network with the best test performance having a number of hiddennodes h, is hence chosen and used with the evaluation data set. The complete list of parameters usedin this work is given in Table 1.

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Start

Processed input signals

h = hmin

N = 1

Train RNN

Is valida-tion errorincreasing ?

Early stopping

Calculatetest error

N = 10?N = N + 1

h = hmax?h = h + dh

Select RNN with lowest test error

End

no

no

yes

yes

yes

yes

no

Figure 7. Flowchart showing the model training and validation procedure.

Table 1. RNN parameters list.

Parameter Value

Model properties Activation function (hidden layer) sigmoidActivation function (output layer) linear

delay, d 2hidden nodes search range, hmin : dh : hmax 4:1:13

number of hidden nodes, h 12training repetitions, N 10

Input No. of play operators 6Training set data 70% of data set

epochs ≤1000algorithm Levenberg-Marquadt

Validation set data 15% of data setTesting set data 15% of data set

Evaluation set data unseen datametric NRMSE

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2.6. Performance Indicator

The performance indicator used in this work to represent the error between the model andexperimental measurements is the normalised root mean square error (NRMSE):

NRMSE(y, y) =1

max(y)−min(y)

√√√√ 1N

N

∑i=1

(yi − y)2i (15)

where y is the actual value, y is the modelled quantity and N is the number of samples considered.By normalising the root mean square error, the errors can be scaled relatively to the range of themeasurement, thus allowing an appropriate comparison across different conditions.

3. Results

Results from the model training, testing and sensitivity analysis are presented and discussed inthis section.

3.1. Minor and Major Loop Model Prediction

Following the model validation and training procedure explained in the previous section, the bestperforming model is used for predicting the data in the evaluation set shown in Figure 6. In order to getrid of the initial transient phase of the model, the first few predicted samples in the set are discarded.In the case of the major loop, a NRMSE of 0.58% is noted in predicting a hysteresis loop with a newramp-rate not used for training. Figure 8 shows the major loop data used for training and evaluatingthe model, as well as the predicted data. For the minor loop data, a NRMSE of 0.66% is noted withdifferent random ramp-rates, including ramp-rates not used for training the model. The magnetic fluxdensity as a function of time is shown in Figure 9.

-6000 -4000 -2000 0 2000 4000 6000

H [Am-1

]

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

B [T

]

Training data

Evaluation data

Predicted data

Figure 8. Predicted major loop and experimental data used in training the model.

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0 5 10 15 20 25 30 35 40 45 50

Time [s]

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

B [T

]

Actual

Estimated

Figure 9. Predicted minor loop with random ramp-rates.

3.2. Effect of Preisach Operators on Performance

In order to quantify the impact of Preisach operators as part of the model, the model’s trainingand validation procedure is repeated without the Pj inputs. Comparing the performance of the twomodels, the inclusion of the Preisach operators is noted to improve performance by 19% in the case ofmajor loop hysteresis, and 44% in the case of minor loop hysteresis. It has to be noted however, that amodel without Preisach operators is computationally less expensive, as the optimal structure contains241 weights versus 409 weights in the original model. Hence, in the end, a compromise must be madebetween accuracy and speed according to the model’s application.

3.3. Univariate Sensitivity Analysis

A univariate sensitivity analysis provides information on how robust a model is when the inputvalues are varied over a specific range [45]. This analysis is performed to understand which modelinput variables impact the prediction most significantly, especially since the magnetic field derivativesignal, H(t) is noisy. A neural network which picks up minor noise during training can overfit thenoise as if it is the signal, leading to poor accuracy during validation [46]. One way to confirm thisis by performing a sensitivity analysis, where one input parameter is fed a changing signal whilstkeeping the other inputs constant, and checking the deviation in the output signal. This is repeated forall the model inputs.

In this exercise, once the model is trained, an input vector is defined and set as 0. Then, for one ofthe eight variables, a value is assigned between the [−1,1] range. This is repeated for 10,000 samplesfor each input variable. The predicted output, ys is saved and the standard deviation σi(ys) for eachvariable i is calculated. The bar chart in Figure 10 illustrates these results. The analysis results showa higher variation in the predicted output for play operator input parameters, thus the model ismore sensitive to changes in these particular variables. These results also confirm the fact that eventhough the model is trained with a noisy H signal, the predicted output is not particularly sensitive toperturbations from this parameter.

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Figure 10. Sensitivity analysis results for each individual input.

4. Conclusions

A Preisach-RNN model is proposed to predict the dynamic characteristics of ferromagneticmaterials that does not require a priori knowledge of the material and its microstructural behaviour.The model is based on the Preisach memory block where the density function is represented by arecurrent neural network. A thorough training and validation procedure is proposed for the neuralnetwork, in order to optimize the weight parameters. We have demonstrated, using ARMCO pureiron measurements, that such a model can predict both major and minor loop dynamic ferromagnetichysteresis behaviour allowing researchers to estimate the dynamic effects of the material knowingonly six different examples at three frequencies. Comparing the model’s predictions to experimentaldata, the model’s NRMSE is noted to be better than 0.7%. Moreover, these results prove that themodel generalises well to new data, and can potentially be used at different frequencies than thefrequencies used in this work. In validating the performance of the model, the positive impact ofPreisach operators is noted, even though this comes at a computational cost, which has to be evaluatedfor each specific industry. Results from a univariate sensitivity analysis also demonstrate that the playoperator inputs impact the predicted magnetic flux output most significantly. We believe that thismodel, predicting major and minor loop hysteresis under different dynamic conditions can be appliedto other dynamic hysteresis prediction problems in the realm of materials science.

Author Contributions: Conceptualization, C.G., M.B., M.P. and N.S.; methodology, C.G. and M.P.; software, C.G.,and M.P.; validation, C.G.; formal analysis, C.G. and M.P.; investigation, C.G. and M.P.; data curation, C.G. andM.P.; writing—original draft preparation, C.G. and M.P.; writing—review and editing, C.G., M.B., M.P. and N.S.;visualization, C.G.; supervision, M.B. and N.S. All authors have read and agreed to the published version ofthe manuscript.

Funding: The publication fee was funded by CERN.

Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

ANN Artificial Neural NetworkCERN European Organization for Nuclear ResearchNRMSE Normalised Root Mean Square ErrorRNN Recurrent Neural Network

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